1. Counting Sample Points
Rule 1: If an operation can be performed in ways, and if for each of these ways a second operation can be performed in ways, then the two operations can be performed together in ways.
Example: Sample points in rolling a pair of dice.
First die can land face up in any one of ways.
For each of these 6 ways the second can also come
up with ways.
So, the two dice can land in (6)(6)=36 possible ways.

2. The sample space S will be of the form:

3. Example : A developer of a new subdivision offers prospective home buyers a choice of Tudor, rustic, colonial, and traditional exterior style in ranch, two story and split level floor plans. In how many different ways a buyer can order one of these homes? Solution: The possible construction ways can be given as

4. Styles offered by Developer
Tudor
Ranch
Two Story
Split Level
Rustic
Colonial
Traditional
Exterior Style
Floor Plan

5. Since and a buyer must choose from possible homes
Since and a buyer must choose from possible homes. Example: If a 22 member club needs to elect a chair and a treasurer, how many different ways can these two be selected? Sol: i) For the chair position, no of possibilities =22 ii) for the treasurer, the remaining possibilities =21 So, both can be selected in (22)(21)=462 different ways.

6. Rule 2: If an operation can be performed in ways, and for each of these a second operation can be performed in
ways, and for each of the first two, a third operation can be performed in ways, and so forth, then the sequence of k operations can be performed in ways.
Example : Sam is going to assemble a computer by himself. He has the choice of
chips from 2 brands,
a hard drive from 4,
Memory from 3 and
An accessory bundle from 5 local stores.
Sam has ways to order the parts.

7. Permutations:
A permutation is an arrangement of all or part of a set of objects.
Example: Consider the 3 letters a, b and c.
The possible permutations are 6 : abc, acb, bac, bca, cab and cba.
The number of choices for 1st letter are
for 2nd letter
for 3rd letter

8. Factorial: For any non negative integer n, called “n factorial” is defined as Theorem: The number of permutations of n objects is n! . Theorem: The number of permutations of n distinct objects taken r at a time is Example: In one year, three awards (research, teaching and service) will be given to a class of 25 graduate students in a statistics department. If each student can receive at most one award, how many possible selections are there?

9. Sol: Since the awards are distinguishable, it is a permutation problem.
The number of possible selections is
_________________
Example: A president and a treasurer are to be chosen from a student club consisting of 50 people. How many different choices of officers are possible if
There are no restrictions;
A will serve only if he is president;
B and C will serve together or not at all?
D and E will not serve together?

10. Sol:
a) The total no of choices of officers, without any restriction is
Since A will serve only if he is a president, we have two situations here:

11. A is selected as a president A is not selected as a president
i) One choice for president,
ii) Remaining 49 outcomes for treasurer.
Officers are selected from remaining 49 people without A.
no of choices=
The total number of choices = =2401

12. B and C are chosen together B and C are not chosen together
c)B and C will serve together
B and C are chosen together
B and C are not chosen together
No of selections when B and C work together =2
i.e (2)(1)=2
If B and C both are not chosen, we have 48 members left.
No of choices=
The total number of choices =2+2256=2258

13. d) D and E will not serve together.
i)D serves as an officer but not E
ii) E serves as an officer but not D
D can take any one of the 2 positions, while the second will be chosen from rest 48 people out of 50 because E is not included,
No of choices= (2)(48)=96
E can be chosen for one of the 2 positions and remaining from 48, as D will not be included.
No of choices=(2)(48)=96
The total number of choices = =2448
iii) Both D and E are not chosen
No of selections =

14. Theorem: The number of permutations of n objects arranged in a circle is (n-1)! . Theorem: The number of distinct permutations of n things of which are of one kind, of a second kind, …, of a kth kind is Example: In a college football training session, the defensive coordinator needs to have 10 player standing in a row. Among these 10 players, there are 1 freshman,2 sophomores, 4 juniors and 3 seniors. How many different ways can they be arranged in a row if only their class level will be distinguished?

15. Sol: no of ways is Partition of a Set: A partition of a set has been achieved if the intersection of every possible pair of r subsets (cells) is the empty set φ and if the union of all subsets gives the original set. Theorem: The number of ways of partitioning a set of n objects into r cells with elements in the first cell, elements in the second, and so forth, is Where

16. Example: In how many ways can 7 graduate students be assigned to 1 triple and 2 double hotel rooms during a conference? Theorem : The number of combinations of n distinct objects taken r at a time is Example: A young boy asks his mother to get 5 Game cartridges from his collection of 10 arcade and 5 sports games. How many ways are there that his mother can get 3 arcade and 2 sports games? Example: How many different letter arrangements can be formed from the letters in the word STATISTICS?

17. 2.29. In a fuel economy study, each of 3 race cars is tested using 5 different brands of gasoline at 7 test sites located in different regions of the country. If 2 drivers are used in the study, and test runs are made once under each distinct set of conditions, how many test runs are needed? 2.38 Four married couples have bought 8 seats in the same row for a concert. In how many different ways can they be seated (a) with no restrictions? (b) if each couple is to sit together? c) If all men sit together to the right of all the women If a multiple-choice test consists of 5 questions, each with 4 possible answers of which only 1 is correct, (a) in how many different ways can a student check off one answer to each question? (b) in how many ways can a student check off one answer to each question and get all the answers wrong?

18. PROBABILITY OF AN EVENT
Definition: The probability of an event A is the sum of the weights of all sample points in A. Therefore,
and
Furthermore if … is a sequence of mutually exclusive events, then
Example : A coin is tossed twice. What is the probability that atleast one head occurs.

19. A die is loaded in such a way that an even number is twice as likely to occur as an odd number. If E is the event that a number less than 4 occurs on a single toss of die, find P(E).

20. Theorem: If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, then the probability of event A is . Example: A statistics class for engineers consists of 25 industrial, 10 mechanical, 10 electrical and 8 civil engineering students. If a person is randomly selected to answer a question, find the probability that the student chosen is a) an industrial engineering major, b) a civil engineering or an electrical engineering major.

21. Additive Rules Theorem: If A and B are two events, then
Corollary 1: If A and B are mutually exclusive events, then

22. Corollary 2: If , , , are mutually exclusive, then Partition of a set: A collection of events of a sample space S is called a partition of sample space if are mutually exclusive and . Corollary3: If is a partition of sample space S, then . An Integer is chosen at random from the first 200 positive integers. What is the probability that the integer chosen is divisible by 6 or 8?

23. Three horses A,B and C are in a race, A is twice as likely to win as B and B is twice as likely to win as C. what is the probability that A or B wins? Solution: P(C)= p P(B)= 2p P(A)= 2P(B)= 4p Since events are mutually and exhaustive so p+2p+4p=1 P(AUB)=6/7

24. Theorem: For three events A, B and C,

25. A card is drawn at random from a deck of ordinary playing cards
A card is drawn at random from a deck of ordinary playing cards. What is the probability that it is a diamond, a face or a king?

26. Theorem: If and are complementary events, then
Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is Suppose also that the probability that the component survives no longer than 4000 hours is 0.04.
(a) What is the probability that the life of the component
is less than or equal to 6000 hours?
(b) What is the probability that the life is greater than
4000 hours?

27. From past experiences a stockbroker believes that under present economic conditions a customer will invest in tax-free bonds with a probability of 0.6, will invest in mutual funds with a probability of 0.3, and will invest in both tax-free and mutual funds with a probability of At this time, find the probability that a customer will invest
In either tax-free bonds or mutual funds;
In neither tax-free bonds nor mutual funds.

28. If 3 books are picked at random from a shelf containing
5 novels, 3 books of poems, and a dictionary, what is the probability that
(a) the dictionary is selected?
(b) 2 novels and 1 book of poems are selected?
Solution: = =

29. Let A be the event that the component fails a particular test and B be the event that the component displays strain but does not actually fail. Event A occurs with probability 0.20, and event B occurs with probability 0.35.
(a) What is the probability that the component does not fail the test?
(b) What is the probability that the component works perfectly well (i.e., neither displays strain nor fails the test)?
(c) What is the probability that the component either fails or shows strain in the test?

30. In addition the worker’s shift 7:00 A.M – 3:00 P.M (day shift),
Factory workers are constantly encouraged to practice zero tolerance when it comes to accidents in factories. Accidents can occur because the working environment or conditions themselves are unsafe. On the other hand, accidents can occur due to carelessness or so-called human error.
In addition the worker’s shift 7:00 A.M – 3:00 P.M (day shift),
3:00 P.M – 11:00 P.M (evening shift)
11:00 P.M – 7:00 A.M (night shift) may be a factor. During the last year 300 accidents have occurred. The percentages of the accidents for the condition combinations are as follows:
If the accident report is selected at random from 300 reports,
What is the probability that the accident occurred on the night shift?
What is the probability that the accident occurred due to human error?
Shift
Unsafe Conditions
Human Error
Day 5%
32%
Evening 6%
25%
Night 2%
30%

31. c) What is the probability that the accident occurred due to unsafe conditions?
What is the probability that the accident occurred on either the
Evening or night shift?

32. It is common in many industrial areas to use a filling machine to fill boxes full of product. This occurs in the food industry as well as other areas in which the product is used in the home, for example,
detergent. These machines are not perfect, and indeed they may A, fill to specification, B, underfill, and C, overfill. Generally, the practice of underfilling is that which one hopes to avoid.
Let P(B) = while P(A) =
(a) Give P(C).
(b) What is the probability that the machine does not
underfill?
(c) What is the probability that the machine either
overfills or underfills?

33. At times, the weight of a product is an important variable to control
At times, the weight of a product is an important variable to control. Specifications are given for the weight of a certain packaged product, and a package is rejected if it is either too light or too heavy. Historical data suggest that 0.95 is the probability that the product meets weight specifications whereas is the probability that the product is too light. For each single packaged product, the manufacturer invests $20.00 in production and the purchase price for the consumer is $25.00.
(a) What is the probability that a package chosen randomly from the production line is too heavy?
(b) For each 10,000 packages sold, what profit is received by the manufacturer if all packages meet weight specification?
(c) Assuming that all defective packages are rejected and rendered worthless, how much is the profit reduced on 10,000 packages due to failure to meet weight specification?